Question

How do you evaluate the expression `tan(u-v)` given `sinu=3/5` with `pi/2<u<p` and `cosv=-5/6` with `pi<v<(3pi)/2`?

How do you evaluate the expression `tan(u-v)` given `sinu=3/5` with `pi/2 < u < pi` and `cosv=-5/6` with `pi < v < (3pi)/2`?

Answer 1

`tan(u-v)=-(432+125sqrt11)/301`
`tan(u-v)~~-2.8126`

Explanation:

Here ,

`sinu=3/5 > 0 ,with , pi/2 < u < pito2^(nd)Quadrant`

`:.cosu=-sqrt(1-sin^2u)=-sqrt(1-9/25)=-sqrt(16/25)`

`:.cosu=-4/5`

`:.tanu=sinu/cosu=((3/5)/(-4/5))`

`:.color(blue)(tanu=-3/4`

Again ,

`cosv=-5/6 < 0,with, pi < v <(3pi)/2to3^(rd)Quadrant`

`:.sinv=-sqrt(1-cos^2v)=-sqrt(1-25/36)=-sqrt11/6`

`:.tanv=sinv/cosv=((-sqrt11/6)/(-5/6))`

`:.color(blue)(tanv=sqrt11/5`

So ,

`tan(u-v)=(tanu-tanv)/(1+tanutanv)`

`color(white)(tan(u-v))=(-3/4-sqrt11/5)/(1+(-3/4)(sqrt11/5)`

`color(white)(tan(u-v))=(-15-4sqrt11)/(20-3sqrt11`

`color(white)(tan(u-v))=(-15-4sqrt11)/(20-3sqrt11 )xx(20+3sqrt11)/(20+3sqrt11)`

`color(white)(tan(u-v))=(-300-45sqrt11-80sqrt11-132)/((20)^2-(3sqrt11)^2)`

`color(white)(tan(u-v))=(-432-125sqrt11)/(400-99)`

`tan(u-v)=-(432+125sqrt11)/301`

`tan(u-v)~~-2.8126`